The Peano Axioms are a good path to retrain math professionals into math teachers. This is because they start at the beginning of counting and go through proving the arithmetic laws such as associativity of addition.
The Peano Axioms teach you how to think about elementary math like a mathematician. They give you the meaning of elementary math. They teach you the concepts behind elementary math.
The Peano Axioms in math foundations books such as Edmund Landau Foundations of Analysis are followed by the math foundations treatment of fractions. Fractions are a continuing problem for many students who have poor fractions understanding many grades later. Obviously, the current system is not working for those students.
Learning fractions in math foundations after the natural numbers gives retrained math types a way to think about explanations of fractions that are based on the actual concept of fractions in math foundations.
Retraining math people into math teaching using the Peano Axioms avoids just saying I know elementary math so I don’t have to learn anything to teach it.
Peano Axioms are fun for adults with a math background to learn.
Search Edmund Landau Foundations of Analysis. This search shows many discussion threads showing this excitement in learning the math foundations explanations of elementary math.
The comment thread at Amazon has comments from adults who have relearned elementary math using the Landau book.
The book Pre-Algebra New Math Done Right Peano Axioms goes also into place value notation as a recursive algorithm, which is not covered in the Edmund Landau book. It also has an easier to understand definition of addition than the Kalmar definition in the Landau book.
The book Pre-Algebra New Axioms Done Right Peano Axioms is now on summer sale at 2.99 for the e-book. E-readers are not needed you can get a free down load ap for computers to read e books from each major vendor for their books.